Calculate the integral using complex analysis
$$\int\limits_{0}^{\infty }\frac{\ln x}{\sqrt[3]{x}(x+8)}dx$$
This integral can easily be calculated using integration by parts, but I was unable to solve it using residues… I don't know which pole should be and which direction
Could you please tell me how to solve this integral using residues?
Best Answer
I used Feymann's trick: $$\text{Let}\quad f\left(x\right)=\int_{0}^{\infty}\frac{t^{x}}{t+8}dt\quad\Rightarrow\quad f'\left(-\frac{1}{3}\right)=\int_{0}^{\infty}\frac{\ln\left(t\right)}{\sqrt[3]{t}\left(t+8\right)}dt$$ $$f(x)=-8^x\pi\csc(\pi x)\quad\Rightarrow\quad f'\left(-\frac{1}{3}\right)=\frac{\pi}{3} (\pi + \sqrt{3} \ln(8))$$
I don't know what you mean by "deductions" but this is a much faster method than integration by parts